The exact worst-case convergence rate of the gradient method with fixed step lengths for L-smooth functions

In this paper, we study the convergence rate of the gradient (or steepest descent) method with fixed step lengths for finding a stationary point of an $L$-smooth function. We establish a new convergence rate, and show that the bound may be exact in some cases, in particular when all step lengths lie in the interval $(0,1\mathrm{/}L]$. In addition, we derive an optimal step length with respect to the new bound.